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September 2008 Perfect trees and elementary embeddings
Sy-David Friedman, Katherine Thompson
J. Symbolic Logic 73(3): 906-918 (September 2008). DOI: 10.2178/jsl/1230396754

Abstract

An important technique in large cardinal set theory is that of extending an elementary embedding j:M→ N between inner models to an elementary embedding j*:M[G]→ N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin’s proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin’s theorem as well as the new result that global domination at inaccessibles (the statement that d(κ) is less than 2κ for inaccessible κ, where d(κ) is the dominating number at κ) is internally consistent, given the existence of 0#.

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Sy-David Friedman. Katherine Thompson. "Perfect trees and elementary embeddings." J. Symbolic Logic 73 (3) 906 - 918, September 2008. https://doi.org/10.2178/jsl/1230396754

Information

Published: September 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1160.03035
MathSciNet: MR2444275
Digital Object Identifier: 10.2178/jsl/1230396754

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.73 • No. 3 • September 2008
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